What is simplify-ln(0.8)+2ln(0.75)-8ln(0.65)+ln(0.6) ?

The solution to simplify-ln(0.8)+2ln(0.75)-8ln(0.65)+ln(0.6) is ln(13.23966…)

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simplify-ln(0.8)+2ln(0.75)-8ln(0.65)+ln(0.6)

simplify

- ln (0.8) + 2 ln (0.75) - 8 ln (0.65) + ln (0.6)

ln (13.23966 \dots)

Decimal Notation

2.58321 \dots

Solution steps

- ln (0.8) + 2 ln (0.75) - 8 ln (0.65) + ln (0.6)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

2 ln (0.75) = ln (0.7 5^{2})

= - ln (0.8) + ln (0.7 5^{2}) - 8 ln (0.65) + ln (0.6)

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (0.7 5^{2}) - ln (0.8) = ln (\frac{0.7 5 ^{2}}{0.8})

= ln (\frac{0.7 5 ^{2}}{0.8}) - 8 ln (0.65) + ln (0.6)

\frac{0.7 5 ^{2}}{0.8} = 0.703125

= ln (0.703125) - 8 ln (0.65) + ln (0.6)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

8 ln (0.65) = ln (0.6 5^{8})

= ln (0.703125) - ln (0.6 5^{8}) + ln (0.6)

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (0.703125) - ln (0.6 5^{8}) = ln (\frac{0.703125}{0.6 5 ^{8}})

= ln (\frac{0.703125}{0.6 5 ^{8}}) + ln (0.6)

\frac{0.703125}{0.6 5 ^{8}} = 22.06610 \dots

= ln (22.06610 \dots) + ln (0.6)

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (22.06610 \dots) + ln (0.6) = ln (0.6 \cdot 22.06610 \dots)

= ln (0.6 \cdot 22.06610 \dots)

Multiply the numbers:

22.06610 \dots \cdot 0.6 = 13.23966 \dots

= ln (13.23966 \dots)

2 lo g_{5} (x) + 4 (lo g_{5} (y) - 2 lo g_{5} (z))

- ln (\frac{C}{V})

ln (2258064 \cdot 1 0^{- 5})

5 lo g_{5} (x) - 25 lo g_{5} (y)

- \frac{5}{4} ln (\frac{1}{4})

What is simplify-ln(0.8)+2ln(0.75)-8ln(0.65)+ln(0.6) ?
The solution to simplify-ln(0.8)+2ln(0.75)-8ln(0.65)+ln(0.6) is ln(13.23966…)